seeing through rock: the mathematics of reflection...
TRANSCRIPT
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Seeing through rock: the mathematics of
Reflection Seismology
William W. Symes
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Reflection seismology is...
• the main prospecting tool of the oil and gas industry
• also useful in civil/environmental engineering and academic earth science
• a $4 billion / year industry
• a big deal in Houston
• rich in mathematical ideas and questions
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The basic idea: collect echoes...
and - through various mathematical operations - convert them to maps of the Earth’sstructure. Infer location of oil and/or gas from structuralmaps, sometimes fromcharacter of echoes also.
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Marine Operations90% of all datacollected worldwide
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Data also acquired on land, but it’s harder...
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Data, data everywhere
Seismic surveys producevastquantities of data!
• each geo/hydro phone collects perhaps 1500 digitized samples per experiment(“shot”).
• each experiment deploys 100’s of geo/hydro phones.
• each survey consists of 100’s (1980’s) to 100,000’s (today)of shots.
• upshot: 750 MB = small, old; 150 GB = modest, contemporary
• equivalent: 1 - 300 CD’s. Immediate future: 100’s of DVD’s!
This data gets turned into...
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An Image of the Earth’s Interior
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(structural image, Mississippi Canyon, offshore Louisiana)
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How do you do that?
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Outline
• Acoustic fundamentals
• Zero offset imaging
• Multioffset imaging
• Velocity via redundancy
• Optimize this!
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Acoustic fundamentals
Basic facts:
• sound ispressurefluctuations about equilibrium - quantify in lb/ft2 or N/m2
• sound isoscillatory- rapid sequence of positive, negative excursions
– A below middle C = 440 cycles/sec (“Hertz”, Hz)
– typical frequency range of significant energy in seismic exploration: 5 - 70Hz
• ⇒ represent pictorially as grey - dark = higher pressure, light = low pressure,plotted vs. time.
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Recording of single hydrophoneNorth Sea survey (thanks: Shell)
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Shot record = recording ofsingle experiment, 64 hydrophonesNorth Sea survey (thanks: Shell)
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CDP 8501
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Acoustic fundamentals
• sound (in rock or in air) is awave phenomenon
• sound travels at a definite speed, characteristic of material
– sound speed in air∼ 330 m/s∼ 1000 ft/s; so count seconds after lightningflash until thunder arrives, multiply by 1000 to get distancein feet to lightningstrike
• reflection seismographs recordechoes⇒ sound must travel fromsound source(dynamite, air gun) to subsurface structureand backto geo/hydro phone.
• echoes (orreflections) occur when sound reaches a change in mechanical char-acteristics (eg. a change in sound speed!) - air to marble, sandstone to limestone,water-saturated sandstone to gassy sandstone,...
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Acoustic fundamentals
Offset= distance from source to receiver. Picture shows “near” and“far” offsets.
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Zero Offset
• source and receiver in same place
• can’t really acquire (dynamite!)
• but can fake it by standard techniques
• easiest to understand how to build image...
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Out and back...
/2vt
v = sound speed (orvelocity) - 330 m/s in air, 1500 m/s in water, 2000 - 5000 m/sin sedimentary rock.
If you hear an echo at timet after seismic wave initiation at zero offset, then thesource of the echo must lie somewhere on a circle of radiusvt/2.
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Finding and old shoe...
here!old shoe lies
Time the echo from the old shoe... “hello!” from each source/receiver point, listenat same place, record arrival time of echo.
Shoe must lie on circles of various radii, computed as in previous slide - thereforeon their intersection!
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However...
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Earth structure does not resemble an old shoe -extended, distributed
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Physics of sound reflection - Zero Offset
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Reflection occurs at the place and time at which the wavefrontis tangentto thereflecting surface - thespecular(mirrorlike) reflection principle. Radius fromsource/receiver point isnormal(perpindicular) to surface.
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Locating a reflecting surface - zero offset
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Reflecting surface must be tangent to each wavefront circle -envelopeof circles.
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Another issue
Do you really want to find thetimes of all the echoes?
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Direct construction of images -ZO Migration
Concept:
(1) for each source/receiver point each sample time t (usually multiples of 2 ms or4 ms),placethe sample value at all possible image points at distancevt/2 (circle).
(2) then sum the resulting images, one for each source/receiver point.
Each image will exhibit circular wavefront.
Principle of Stationary Phase:Sum of oscillatory signalscancels, exceptalongenvelope of wavefronts.
Consequence: after the summation,only the envelope is visible!!!.
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ZO Migration Example (1/4)
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Zero offset simulated data for reflecting surface at depthz = 0.3 km at left edge(x = 0 km), sloping upwards to the right at 5 deg. 101 source/receiver points,spaced 10 m apart, from left to right.
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ZO Migration Example (2/4)
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(1) for each source/receiver point each sample time t (usually multiples of 2 ms or4 ms),placethe sample value at all possible image points at distancevt/2 (circle).
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ZO Migration Example (3/4)
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(2) then sum the resulting images, one for each source/receiver point (this figureshows 5 of the 101 source/receiver images added up, at 0.3, 0.4, 0.5, 0.6, 0.7 kmfrom left edge).
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ZO Migration Example (4/4)
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(2) then sum the resulting images, one for each source/receiver point (this figureshows all 101 source/receiver images added up - the wavefront envelope is recov-ered, sitting exactly on the reflecting surface!)
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Multioffset Migration
Multiple Offsets= data that’s actually measured. Can eitherfake ZO data frommultioffset data (1970’s technology) or directly use MO data - routine today.
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Physics of sound reflection - Nonzero Offset
Recvr
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� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
Source
Reflection occurs at the place and timet at which ellipse with focii at source, recvr,diam =vt is tangentto the reflecting surface - thespecular(mirrorlike) reflectionprinciple. Bisector of lines from source and receiver points isnormal (perpindicu-lar) to surface (Snell’s law) ⇒ surface isenvelopeof ellipses.
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Direct construction of images -MO Migration
Concept:
(1) for each source/receiver point each sample time t (usually multiples of 2 ms or4 ms),placethe sample value at all possible image points on ellipse of diametervt.
(2) then sum the resulting images, one for each source/receiver pair.
Each image will exhibit elliptical wavefronts.
Principle of Stationary Phase:Sum of oscillatory signalscancels, exceptalongenvelope of wavefronts.
Consequence: after the summation,only the envelope is visible!!!.
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And that’s how we turned lots of this...
0
1
2
3
4
5
time
(s)
-4 -3 -2 -1offset (km)
(one of 500 data panels from Exxon GofM data = 750 MB.)
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into this!
0
500
1000
1500
2000
2500
Dept
h in
Met
ers
0 200 400 600 800 1000 1200 1400CDP
(structural image, Mississippi Canyon, offshore Louisiana)
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Two major details
(1) Q. how do you knowv? A. You don’t!
(2) Spatial variation of velocity and refraction of waves (some other time...)
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Velocity via redundancy (1/6)
• Typical survey contains source-receiver pairs with a variety of offsets.
• Any one set of ellipses with fixed offset (source-receiver distance) is enough todetermine reflecting surfaces by forming envelopes -common offsetconstruction
• the envelopes constructed using different offsets will only agree if the velocityis correct!
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Velocity via redundancy (2/6)
Black circle, ellipse: zero, nonzero offset, correct velocity - both have dashed blackline as possible envelope.
Blue circle, red ellipse: same offsets, velocity 50% high. Envelopes aredifferent,so predicted depths of reflecting surface atsameposition will disagree.
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Velocity via redundancy (3/6)
You can see this with the data directly:
• sort the data recordings into piles, each pile (“gather”) sharing common offset;
• carry out migration for each common offset gather separately: place data valueat timet on ellipse of diametervt, sum together to getcommon offset image(justlike we did for zero offset - a special case!);
• from each image, extract the part of the image that lies belowa specific point onthe surface - a “virtual well”;
• if the velocity is correct, then all “virtual wells” will look same! To compare,put them together, one for each offset, to form acommon image gather. Look atit!
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Velocity via redundancy (4/6)
,
0
0.2
0.4
0.6
0.8
1.0
time
(s)
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4offset (km)
Synthetic experiments - data from different offsets
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Velocity via redundancy (5/6)
0
0.2
0.4
0.6
0.8
1.0
dept
h (k
m)
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4offset (km)
,
0
0.2
0.4
0.6
0.8
dept
h (k
m)
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4offset (km)
“Virtual wells” displayed next to each other: Left, correctv; Right, 10% low.
“Correct velocity flattens common image gathers” - the practical principle for find-ing sound velocity from seismic data.
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Velocity via redundancy (6/6)
Works with actual data!
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Optimize This!
If you can measure “flatness” somehow, then you can turn image-gather-flatteningtask into optimization problem:minimize a measure of un-flatness of image gathers
An obvious possibility, apr̀es calculus: measure un-flatness by mean square ofderivative in offset of image gathers≡ thedifferential semblanceobjective (func-tion of assumed velocity, data).
Stolk & WWS Inverse Problems03: differential semblance issmooth in velocityand data jointlyandis essentially the only quadratic form in the image gather withthis property - so the only one for which you can use Newton’s method! [theory ofoscillatory integral operators]
Generally: provided a strong enough dichotomy between length scale in velocitymodel and wavelengths in data, can show local uniqueness of global minimizer. Insome special cases,all local minimizers are global![multiscale asymptotics]
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Some Obvious Questions
• what does all of this have to do with physics?
• waves obey the wave equation (acoustic, elastic, whatever)- where is the waveequation?
• what the heck is an image, really? what does it have to do with the mechanics ofrocks? why is it separate, apparently, from “the velocity”?
• I thought you did inverse problems - is there an inverse problem somewhere inhere?
• why not “image” the earth’s structure by modeling the data and fitting the modelpredictions to the data, say by least squares?
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Where to Find Some Answers
• CAAM Colloquium 13 September
• CIME Short Course,www.trip.caam.rice.edu > Downloadable Materi-als
• Mathematical Foundations of Reflection Seismology, ditto
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Credit where credit is due
Photos, drawings: Landmark Graphics, Schlumberger, Western Geco, Veritas DGC,Seismic Consultants, and WWS
Seismic data: Exxon Production Research Co., Shell International Research
Software used to produce data-derived graphics: Seismic Unix (Colorado Schoolof Mines), The Rice Inversion Project (Rice University)
Money: sponsors of The Rice Inversion Project, NSF, DoE
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